Yes mice can pass on nasty diseases to hamsters. One in particular - can't remember the name. So you need to protect the hamster. Is your hamster in a tank or a cage? Mice can get through bars bigger than 7mm. I had a similar issue at our last house - we were always getting mice. I had to have the hamster in a cage with 7mm bar spacing.
This may not sound popular but I think you need to kill the mice, not catch them. They breed fast and there is never just one mouse. With mousetraps you will probably catch quite a few. We used two types that did the trick. Both of which would also kill a hamster so you need to make sure your hamster cage is very secure or move it to another room.
One was an electronic trap that was quite expensive. But these worked very well. The mice die either in or very close to the trap which means you know where they are and can remove them (rather than them dying underneath something and smelling or being found by another pet and eaten).
Mice are immune to most poisons now, but not this one. It sedates them instantly and they die under sedation so it's perhaps less cruel than most other mice traps. The poison is well enclosed inside the plastic box and is solid so it can't fall out or be touched so safer if there are children around. The entrance hole is not that big but it's possible a hamster could get in it so important to keep the hamster safe.
However within a few days you should have caught and disposed of a family or two of mice and everything be safe again.
You get two of these traps in a box. I would get two packs (ie four traps) - they sit up against a skirting board so put them on the route you think the mice have taken.
I tried all kinds of mouse traps including the humane ones (which aren't that humane as they sometimes die of fright before you set them free). The traditional little nipper ones were truly grim and not that humane either. So we settled for these - quick and simple. It is not pleasant finding dead mice but better than a dead hamster. They're wild mice so carry diseases. And they chew through electrical wires in the wall as well!
https://www.amazon.co.uk/Rentokil-Ac...s%2C164&sr=8-2